<p>This paper is concerned with the propagation dynamics for a large class of nonautonomous cooperative systems with nonlocal dispersal and a time-periodic shifting environment. Under the assumption that each of the two limiting systems has both leftward and rightward spreading speeds, we establish the upward convergence of solutions for such a system by appealing to the abstract theory developed for monotone evolution systems with asymptotic translation invariance. Then, we obtain the asymptotic annihilation by an ingenious approximation. Due to the lack of compactness, the existence of time-periodic forced traveling waves is proved with the aid of the Kuratowski measure of noncompactness. We further prove the uniqueness and global attractivity of the forced wave via a dynamical system approach. Finally, we utilize our stability results to derive the nonexistence of forced wave fronts.</p>